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In music
Music

Music is an art form whose media is sound organized in time. Common elements of music are pitch , rhythm , dynamics , and the sonic qualities of timbre and texture ....
 the axis system, proposed by Erno Lendvai
Erno Lendvai

Erno Lendvai was one of the first theorists to write on the appearance of the golden section and Fibonacci series and how these are implemented in B?la Bart?k's music....
 (1971, p.1-16) in his analysis of the use of tonality
Tonality

Tonality is a system of music in which specific hierarchy pitch relationships are based on a Key "center" or Tonic . The term tonalit? originated with Alexandre-?tienne Choron and was borrowed by Fran?ois-Joseph F?tis in 1840 ....
 in the music of Béla Bartók
Béla Bartók

B?la Viktor J?nos Bart?k was a Hungarian people composer and pianist, considered to be one of the greatest composers of the 20th century. Through his collection and analytical study of folk music, he was one of the founders of ethnomusicology....
, is an assignment of harmonic function to all twelve pitch class
Pitch class

In music, a pitch class is a set of all Pitch that are a whole number of octaves apart, e.g. the pitch class C consists of the Cs in all octaves....
es in relation to an assigned tonic
Tonic (music)

The tonic is the first note of a scale in the tonality method of musical composition. The chord #The Triad formed on the tonic note, the tonic chord, is thus the most significant chord ....
, determined by that pitch class's interval
Interval (music)

In music theory, the term interval describes the relationship between the pitch of two notes.Intervals may be described as:*vertical if the two notes sound simultaneously...
 from the tonic. The three functions used are tonic, subdominant
Subdominant

In music, the subdominant is the technical name for the fourth tonal degree of the diatonic scale. It is so called because it is the same distance "below" the Tonic as the dominant is above the tonic - in other words, the tonic is the dominant of the subdominant....
, and dominant
Dominant (music)

In music, the dominant is the fifth degree of the Scale . For example, in the C major scale , the dominant is the note G; and the dominant chord uses the notes G, B, and D....
, and each contains four notes (forming three diminished seventh chord
Diminished seventh chord

A diminished seventh chord is a four note chord comprising a diminished chord plus the interval of a diminished seventh above the root . Thus it is , or enharmonically , of any major scale; for example, C diminished-seventh would be , or enharmonically ....
s). Each group is then divided into principal and secondary groups a tritone
Tritone

The tritone is a musical interval that spans three major second. The tritone is the same as an augmented fourth, which in equal temperament is enharmonic to a diminished fifth....
 apart.






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In music
Music

Music is an art form whose media is sound organized in time. Common elements of music are pitch , rhythm , dynamics , and the sonic qualities of timbre and texture ....
 the axis system, proposed by Erno Lendvai
Erno Lendvai

Erno Lendvai was one of the first theorists to write on the appearance of the golden section and Fibonacci series and how these are implemented in B?la Bart?k's music....
 (1971, p.1-16) in his analysis of the use of tonality
Tonality

Tonality is a system of music in which specific hierarchy pitch relationships are based on a Key "center" or Tonic . The term tonalit? originated with Alexandre-?tienne Choron and was borrowed by Fran?ois-Joseph F?tis in 1840 ....
 in the music of Béla Bartók
Béla Bartók

B?la Viktor J?nos Bart?k was a Hungarian people composer and pianist, considered to be one of the greatest composers of the 20th century. Through his collection and analytical study of folk music, he was one of the founders of ethnomusicology....
, is an assignment of harmonic function to all twelve pitch class
Pitch class

In music, a pitch class is a set of all Pitch that are a whole number of octaves apart, e.g. the pitch class C consists of the Cs in all octaves....
es in relation to an assigned tonic
Tonic (music)

The tonic is the first note of a scale in the tonality method of musical composition. The chord #The Triad formed on the tonic note, the tonic chord, is thus the most significant chord ....
, determined by that pitch class's interval
Interval (music)

In music theory, the term interval describes the relationship between the pitch of two notes.Intervals may be described as:*vertical if the two notes sound simultaneously...
 from the tonic. The three functions used are tonic, subdominant
Subdominant

In music, the subdominant is the technical name for the fourth tonal degree of the diatonic scale. It is so called because it is the same distance "below" the Tonic as the dominant is above the tonic - in other words, the tonic is the dominant of the subdominant....
, and dominant
Dominant (music)

In music, the dominant is the fifth degree of the Scale . For example, in the C major scale , the dominant is the note G; and the dominant chord uses the notes G, B, and D....
, and each contains four notes (forming three diminished seventh chord
Diminished seventh chord

A diminished seventh chord is a four note chord comprising a diminished chord plus the interval of a diminished seventh above the root . Thus it is , or enharmonically , of any major scale; for example, C diminished-seventh would be , or enharmonically ....
s). Each group is then divided into principal and secondary groups a tritone
Tritone

The tritone is a musical interval that spans three major second. The tritone is the same as an augmented fourth, which in equal temperament is enharmonic to a diminished fifth....
 apart. Thus there are six pairs of notes in three categories.

If the tonic is C, then the functions are as follows:

Functions are assigned based on position within the circle of fifths
Circle of fifths

In music theory, the circle of fifths shows the relationships among the twelve tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys....
, starting with F as subdominant: F (s), C (t), G (d), D (s), etc.

The system is critiqued by Paul Wilson
Paul Wilson (music theorist)

Paul Wilson is a Music theory and Professor of Music Theory and Musical composition at the University of Miami Frost School of Music, in the United States....
 (1992, p.203-208).

Harmonic properties

The axis system shares the "essential properties of classical harmony", i.e.
  • the functional affinities of the fourth and fifth degrees
  • the relationship of relative major and minor keys
  • the overtone relations
  • the role of leading notes
  • the opposite tension of the dominant and the subdominant
  • the duality of tonal and distance principles


Components

The axis system consists of the tonic, dominant, and subdominant axes, each of which consist of a principal and secondary branch, each of which consist of a pole and counterpole (Lendvai 1979, p.5).

If the pole of the principal branch is C then its counterpole is F, together they comprise the principal branch while the secondary branch of the same axis consists of the pole A and the counterpole E. The system is completed by the subdominant and dominant axes, F-D-B-A and G-E-C-A (ibid, p.3). The relation between counterpoles is much more sensitive than the relation between the other poles of an axis, so much so that "a pole is always interchangeable with its counterpole without any change in its function." (ibid, p.4)

"Thus each axis has a two-fold affinity depending on whether we oppose the pole with the counterpole, or the principal branch with the secondary branch." (ibid, p.5) The form of the first movement of Music for Strings, Percussion and Celesta
Music for Strings, Percussion and Celesta

Music for Strings, Percussion and Celesta Sz. 106, BB 114 is one of the best-known Musical composition by the Hungary composer B?la Bart?k. Commissioned by Paul Sacher to celebrate the tenth anniversary of the Basel Chamber Orchestra, the score is dated September 7, 1936....
 illustrates this, where the opening is based upon the pole of the principal branch of the tonic axis (A, A-C-E-F), the middle contrasts the opening and the end of the first movement by being based on E. In the second movement the beginning and ending based on C are contrasted with the middle based on its pole F. The two movements themselves are contrasted through the first being based on the principal branch (A-E) and the second on the secondary (C-F).

It is interesting to note that the pattern Tonic-Dominant-Subdominant (T-D-S) repeats itself around the circle of fifths (ibid, p.2): C G D A E B F D A E B F T D S T D S T D S T D S

Composition and analysis with the axis system


It may be noted that each of the above notes forms a descending diminished chord. In the case of I or C, C-A-F-D, or an E fully diminished chord. This connects the axis system not only with diminished chords, which often form the basis for the movement in a piece based on the axis system, but also links the axis system with the diminished scales formed on III of each of the principal tones: root, subdominant, and dominant.

The axis system is probably used more in analysis, and less in composition. Uses of the system have a characteristic sound, but that sound is similar to that which can be found in uses of tritone Dominant Substitutions, and Deceptive Cadences using the VII dominant chord. Older systems of harmonic theory segue very easily into the axis system. It might be noted that many composers, like Bartók, are extremely "mathematical" while many are not.

The axis system and jazz

Brazilian music, especially the music of Antonio Carlos Jobim
Antônio Carlos Jobim

Ant?nio Carlos Brasileiro de Almeida Jobim , also known as Tom Jobim, was a Grammy Award-winning Brazilian songwriter, composer, arranger, singer, and pianist/guitarist....
, uses harmonies that are easily analyzed by the axis system and not easily analyzed in other ways. A great example is the bridge of The Girl from Ipanema
The Girl from Ipanema

"The Girl from Ipanema" is a well-known bossa nova song, a worldwide hit in the mid-1960s that won a Grammy Award for Grammy Awards of 1965. It was written in 1962, with music by Antonio Carlos Jobim and Portuguese lyrics by Vinicius de Moraes with English language lyrics written later by Norman Gimbel....
.

See also

  • Acoustic scale
    Acoustic scale

    In music, the acoustic scale is a seven note Scale which, starting on C, contains the notes: C, D, E, F, G, A and B. This differs from the major scale in having a sharp fourth and flat seventh scale degree....
  • Coltrane changes
    Coltrane changes

    In jazz harmony, the Coltrane changes are a chord progression variation using substitute chords over common jazz chord progressions. These substitution patterns were first demonstrated by jazz musician John Coltrane on the album Blue Train_ on the tunes Lazy Bird and Moment's Notice....


Source

Wilson, Paul (1992). The Music of Béla Bartók. ISBN 0-300-05111-5.